Scientific Notation (exponential notation) Chemistry Tutorial

Key Concepts

• Scientific notation is also referred to as exponentional notation.
• Scientific notation (exponential notation) is a convenient way to write down a very large or a very small number.
• In scientific notation each number is written as a product of two numbers, that is, two numbers are multiplied together:

  a coefficient × 10^{an exponent}

  1.234 × 10⁵

• Coefficients are usually expressed with one digit to the left of the decimal point.⁽¹⁾

  A coefficient is expressed as a number between 1 and 10. ⁽²⁾

• An exponent gives the position of the decimal point in the number and is either:

  ⚛ positive, in which case the + sign is omitted

  (generally for numbers greater than or equal to 10)
  10 = 1 × 10¹
  900 = 9 × 10²

  ⚛ zero

  (generally for numbers between 1 and 10)
  1 = 1 × 10⁰
  9 = 9 × 10⁰

  ⚛ negative, and the minus sign (-) is written to the left of the value of the exponent

  (generally for numbers between 0 and 1)
  0.1 = 1 × 10^-1
  0.09 = 9 × 10^-2

  Exponent	negative (-)	0	positive
  Number	0 up to 1	between 1 and 10	10 or greater
  Example	0.12 = 1.2 × 10-1	1.2 = 1.2 × 100	12 = 1.2 × 101

• Convert a number given as a decimal system number to a number in scientific notation by:

  (1) Writing the coefficient

  coefficient is between 1 and 10

  (2) Determining the sign of the exponent based on the size of the number:

  0 < number < 1	exponent is negative (-)
  1 < number < 10	exponent is 0
  number ≥ 10	exponent is positive

  (3) Determining the value of the exponent based on how many places the decimal point has moved between its position in the decimal number and its positon in the coefficient

  (4) Writing the number in the form of: coefficient × 10^exponent

• Convert a number given in scientific notation to a number in the decimal system by:

  (1) Writing the coefficent

  (2) Deciding which way to move the decimal point based on the sign of the exponent:

  negative exponent: move decimal point left

  positive exponent: move decimal point right

  (3) Deciding how many places to move the decimal point based on the magnitude (size) of the exponent

  (4) Move the decimal point in the coefficient the correct number of places in the correct direction.

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Big Numbers and Small Numbers In Scientific Notation (exponential notation)

Chemists regularly use some pretty big numbers, some examples are given below:

• 602000000000000000000000, is known as Avogadro's Constant
• 101300 is atmospheric pressure in pascals (Pa)
• 96484 is known as the Faraday Constant
• 300000000 which is the speed of light in a vacuum in m s^-1

Imagine trying to do calculations involving Avogadro's Constant, 602000000000000000000000, it is really hard to keep track of all those zeroes!
If you accidently leave off just one of those zeroes then the number you write will be 10 times smaller than it should have been.
If you lost two zeroes, then the number would be 100, times smaller than it should be!
If you lost three zeroes, then the number would be 1000 times smaller than it should be ....
and are you starting to see a pattern here?

602000000000000000000000 could be thought of as the product of a number multiplied by 10, or by 10 × 10 (100), or by 10 × 10 × 10 (1000), etc as shown below:

• 602,000,000,000,000,000,000,000 = 60,200,000,000,000,000,000,000 × 10
• 602,000,000,000,000,000,000,000 = 6,020,000,000,000,000,000,000 × 10 × 10
• 602,000,000,000,000,000,000,000 = 602,000,000,000,000,000,000 × 10 × 10 × 10
• 602,000,000,000,000,000,000,000 = 60,200,000,000,000,000,000 × 10 × 10 × 10 × 10
• 602,000,000,000,000,000,000,000 = 6,200,000,000,000,000,00 × 10 × 10 × 10 × 10 × 10

Now if we had a short-hand way of representing all those "× 10s" we would have a useful way to keep track of all those zeroes.

Well mathematicians have developed this short-hand. They use a small superscript number to the right of the number 10 to tell us how many "× 10"s are needed to make up the number.
We are going to call this number the exponent.

The number "10" is just 1 "× 10" so the exponent is "1": 10¹, but usually we just write 10.

The number "100" is 1 "× 10 × 10", there are, "2" "×10s" so the exponent is "2": 10².

The number "1,000" is 1 "× 10 × 10 × 10", that is, "3" "×10s" so the exponent is "3": 10³.

The number "10,000" is 1 "× 10 × 10 × 10 × 10", that is, "4" "×10s" so the exponent is "4": 10⁴.

Do you see another pattern emerging?
We are replacing each "0" in the number with "×10" and the exponent increases by "1".
So we could just count the number of "0"s after the last non-zero digit in our original number to determine the value of the exponent.

So we could represent Avogadro's Constant, 602,000,000,000,000,000,000,000, as 602 with 21 "0"s after it as:

602 × 10²¹

602 × 10²¹ is much easier to write than 602,000,000,000,000,000,000,000 but we can keep going as shown below:

602 × 10²¹ = 60.2 × 10 × 10²¹ = 60.2 × 10²²

60.2 × 10²² = 6.02 × 10 × 10²² = 6.02 × 10²³

We usually express Avogadro's Constant as 6.02 × 10²³.
When we represent numbers as small number between 0 and 10 (6.02) multiplied by 10 to the power of another number (× 10²³) we call this scientific notation or exponential notation.
The small number between 0 and 10 (6.02) is called the coefficient.
The "power" we raise the "10" to is called the exponent (23 in this case).

Chemists also work with some really small numbers. For example:

• the mass of an electron is 0.0000000000000000000000000000009109 kilograms
• the mass of a proton is 0.000000000000000000000000001673 kilograms
• an atom has a diameter of about 0.0000000002 metres

In these cases we divide the number by 10, let's take the diameter of an atom as an exmple:

• 0.0000000002 = 0.000000002 ÷ 10
• 0.0000000002 = 0.00000002 ÷ (10 × 10)
• 0.0000000002 = 0.0000002 ÷ (10 × 10 × 10)
• 0.0000000002 = 2 ÷ (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10)

Which, using what we learnt above about exponents, we could also write as:

• 0.0000000002 = 0.000000002 ÷ 10 = 0.000000002 ÷ 10¹
• 0.0000000002 = 0.00000002 ÷ (10 × 10) = 0.00000002 ÷ 10²
• 0.0000000002 = 0.0000002 ÷ (10 × 10 × 10) = 0.0000002 ÷ 10³
• 0.0000000002 = 2 ÷ (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10) = 2 ÷ 10¹⁰

We don't usually express numbers as a division by 10 raised to some power, instead we write the number in scientific notation using multiplication of 10 raised to some power.
How can you convert a division to a muliplication?

Mathematicians have a trick for this too!
They use a minus sign (-) before the first number of the exponent.
Some examples are shown below:

• 0.000000002 ÷ 10¹ = 0.000000002 × 10^-1
• 0.00000002 ÷ 10² = 0.00000002 × 10^-2
• 0.0000002 ÷ 10³ = 0.0000002 × 10^-3
• 2 ÷ 10¹⁰ = 2 × 10^-10

So, using scientific notation (exponential notation), the diameter of an atom is 2 × 10^-10 metres.
The coefficient is 2
The exponent is -10

To summarise, scientific notation uses a small number (between 0 and 10) as the coefficient and multiplies that number by 10 raised to the power of a number called the exponent.

coefficient × 10^exponent

If the number is greater than or equal to 10, the exponent will be positive (but we don't write the + sign)

coefficient × 10^exponent

If the number is between 1 and 10, the exponent will be zero (0)

coefficient × 10⁰

If the number is between 0 and 1, the exponent will be negative (we DO write the - sign before the value of the exponent)

coefficient × 10^-exponent

Sometimes you might need to convert a number given in scientific notation (exponential notation) to a decimal system number.
For example, you might need to weigh out 1.325 × 10³ grams of table salt (sodium chloride).
The digital display on your electronic balance only gives numbers in the decimal system, so you will need to convert 1.325 × 10³ to a decimal system number.
Because the exponent is positive (no minus sign before the value of the exponent) we know we need to multiply "1.325" by "3" "× 10"s, that is:

1.325 × 10 × 10 × 10 = 1325

In effect, we are moving the decimal point to the right. The number of places the decimal point moves is equal to the value of the exponent:

1.325 × 10³ = 1325. = 1325

What if we needed to weigh out 7.6 × 10^-3 grams of table salt (sodium chloride)?
Because the exponent is negative (a minus sign before the value of the exponent) we know we need to divide 7.6 by 10³, or divide 7.6 by "3" "× 10"s, that is:

7.6 × 10^-3 = 7.6 ÷ 10³ = 7.6 ÷ (10 × 10 × 10) = 7.6 ÷ 1000 = 0.0076

In this case, we are moving the decimal point in the coefficient to the left. The number of places it moves is equal to the magnitude (size) of the exponent. If there is no number in the position where the decimal point moves, we insert a "0"

7.6 × 10^-3 = .0076 = 0.0076

Converting a Number to Scientific Notation (exponential notation)

We can write any number using scientific notation (exponential notation) if we follow a few simple steps:

1. Locate the digits of the coefficient (first through to last non-zero numbers⁽¹⁾)
2. Convert the digits of the coefficient to a coefficient, a number between 0 and 10 ⁽²⁾

   coefficient = ?

3. Count the number of "×10"s (place holders) between the decimal point in the coefficient and the decimal point in the original number.
   This will be the magnitude (size) of the exponent.

   The magnitude (or size) of the exponent is ?

4. Determine the sign of the exponent:

   If the original number is 10 or more, the exponent is positive (DO NOT include the + sign)

   If the original number is between than 1 and 10, the exponent is 0

   If the original number is between 0 and 1, the exponent is negative (DO include the - sign)

   exponent = ?

5. Write the number in the form: coefficient × 10^exponent

Worked Examples of Writing Numbers Using Scientific Notation (exponential notation)

Question 1: Write the number 0.015 using scientific notation (exponential notation).

Solution:

1. Locate the digits of the coefficient (first through to last non-zero numbers)

   0.015

2. Convert the digits of the coefficient to a coefficient, a number between 0 and 10

   15 becomes 1.5

3. Count the number of "×10"s (place holders) between the decimal point in the coefficient and the decimal point in the original number.

   Locate decimal points		↓			↓
   Number to convert	0	.	0	1	.	5
   Number of "× 10"s		↑	← (2)	← (1)	↑

   The magnitude (or size) of the exponent is 2

4. Determine the sign of the exponent:

   If the original number is 10 or more, the exponent is positive (DO NOT include the + sign)

   If the original number is between than 1 and 10, the exponent is 0

   If the original number is between 0 and 1, the exponent is negative (DO include the - sign)

   0.015 < 0

   exponent = -2

5. Write the number in the form: coefficient × 10^exponent

   0.015 = 1.5 × 10^-2

Question 2: Write the number 256.35 using scientific notation (exponential notation).

Solution:

1. Locate the digits of the coefficient (first through to last non-zero numbers)

   256.35

2. Convert the digits of the coefficient to a coefficient, a number between 0 and 10

   256.35 becomes 2.5635

3. Count the number of "×10"s (place holders) between the decimal point in the coefficient and the decimal point in the original number.

   Locate decimal points		↓			↓
   Number to convert	2	.	5	6	.	3	5
   Number of "× 10"s		↑	→ (1)	→ (2)	↑

   The magnitude (or size) of the exponent is 2

4. Determine the sign of the exponent:

   If the original number is 10 or more, the exponent is positive (DO NOT include the + sign)

   256.35 > 10

   exponent = 2

   If the original number is between than 1 and 10, the exponent is 0

   If the original number is between 0 and 1, the exponent is negative (DO include the - sign)

5. Write the number in the form: coefficient × 10^exponent

   256.35 = 2.5635 × 10²

Question 3: Write the number 42.06 using scientific notation (exponential notation).

Solution:

1. Locate the digits of the coefficient (first through to last non-zero numbers)

   42.06

2. Convert the digits of the coefficient to a coefficient, a number between 0 and 10

   42.06 becomes 4.206

3. Count the number of "×10"s (place holders) between the decimal point in the coefficient and the decimal point in the original number.

   Locate decimal points		↓		↓
   Number to convert	4	.	2	.	0	6
   Number of "× 10"s		↑	→ (1)	↑

   The magnitude (or size) of the exponent is 1

4. Determine the sign of the exponent:

   If the original number is 10 or more, the exponent is positive (DO NOT include the + sign)

   42.06 > 10

   exponent = 1

   If the original number is between than 1 and 10, the exponent is 0

   If the original number is between 0 and 1, the exponent is negative (DO include the - sign)

5. Write the number in the form: coefficient × 10^exponent

   42.06 = 4.206 × 10¹

Question 4: Write the number 3.56 using scientific notation (exponential notation).

Solution:

1. Locate the digits of the coefficient (first through to last non-zero numbers)

   3.56

2. Convert the digits of the coefficient to a coefficient, a number between 0 and 10

   3.56 remains 3.56

3. Count the number of "×10"s (place holders) between the decimal point in the coefficient and the decimal point in the original number.

   Locate decimal points		↓
   Number to convert	3	.	5	6
   Number of "× 10"s		(0)

   The magnitude (or size) of the exponent is 0

4. Determine the sign of the exponent:

   If the original number is 10 or more, the exponent is positive (DO NOT include the + sign)

   If the original number is between than 1 and 10, the exponent is 0

   1 < 3.56 < 10

   exponent = 0

   If the original number is between 0 and 1, the exponent is negative (DO include the - sign)

5. Write the number in the form: coefficient × 10^exponent

   3.56 = 3.56 × 10⁰

Converting Scientific Notation to a Decimal System Number

Use the following steps to convert a number given in scientific notation (exponential notation) to a decimal system number:

1. Write the coefficient
2. Decide which way to move the decimal point in the coefficient:

   Positive exponent: move the decimal point to the right

   Exponent = 0: do not move the decimal point

   Negative exponent: move the decimal point to the left

3. Decide how many places to move the decimal point based on the magnitude (or size) of the exponent
4. Move the decimal point in the coefficient the correct number of places in the correct direction using "0"s to fill any places not occupied by a non-zero digit.

Worked Examples of Converting Numbers in Scientific Notation to Decimal System Numbers

Question 1: Write 1.23 × 10³ as a decimal system number.

Solution:

1. Write the coefficient

1.23

2. Decide which way to move the decimal point in the coefficient:

   Positive exponent: move the decimal point to the right

   exponent = 3 therefore positive (no negative sign)

   Exponent = 0: do not move the decimal point

   Negative exponent: move the decimal point to the left

3. Decide how many places to move the decimal point based on the magnitude (or size) of the exponent

   exponent = 3 so move decimal point 3 places to the right

4. Move the decimal point in the coefficient the correct number of places in the correct direction using "0"s to fill any places not occupied by a non-zero digit.

   decimal point in coefficient		↓
   coefficient	1	.	2	3
   move decimal point 3 places to the right		↑	→ (1)	→ (2)	→ (3)	↓
   insert zeroes where required	1		2	3	0	.

   1.23 × 10³ = 1230

Question 2: Write 4.76 × 10^-2 as a decimal system number.

Solution:

1. Write the coefficient

4.76

2. Decide which way to move the decimal point in the coefficient:

   Positive exponent: move the decimal point to the right

   Exponent = 0: do not move the decimal point

   Negative exponent: move the decimal point to the left

   exponent = -2 therefore negative (a minus sign)

3. Decide how many places to move the decimal point based on the magnitude (or size) of the exponent

   exponent = -2 so move decimal point 2 places to the left

4. Move the decimal point in the coefficient the correct number of places in the correct direction using "0"s to fill any places not occupied by a non-zero digit.

   decimal point in coefficient					↓
   coefficient				4	.	7	6
   move decimal point 2 places to the left		↓	← (2)	← (1)	↑
   insert zeroes where required		.	0	4		7	6

   4.76 × 10^-2 = .0476 = 0.0476

Question 3: Write 5.22 × 10¹ as a decimal system number.

Solution:

1. Write the coefficient

5.22

2. Decide which way to move the decimal point in the coefficient:

   Positive exponent: move the decimal point to the right

   exponent = 1 therefore positive (no negative sign)

   Exponent = 0: do not move the decimal point

   Negative exponent: move the decimal point to the left

3. Decide how many places to move the decimal point based on the magnitude (or size) of the exponent

   exponent = 1 so move decimal point 1 place to the right

4. Move the decimal point in the coefficient the correct number of places in the correct direction using "0"s to fill any places not occupied by a non-zero digit.

   decimal point in coefficient		↓
   coefficient	5	.	2	2
   move decimal point 1 place to the right		↑	→ (1)	↓
   insert zeroes where required	5		2	.		2

   5.22 × 10¹ = 52.2

Question 4: Write 9.45 × 10⁰ as a decimal system number.

Solution:

1. Write the coefficient

9.45

2. Decide which way to move the decimal point in the coefficient:

   Positive exponent: move the decimal point to the right

   Exponent = 0: do not move the decimal point

   exponent = 0 therefore we DO NOT move the decimal point

   Negative exponent: move the decimal point to the left

3. Decide how many places to move the decimal point based on the magnitude (or size) of the exponent

   exponent = 0 so DO NOT move the decimal point

4. Move the decimal point in the coefficient the correct number of places in the correct direction using "0"s to fill any places not occupied by a non-zero digit.

   9.45 × 10⁰ = 9.45